Question

In the adjoining figure, two circles intersect each other at points $S$ and $R$. Their common tangent $PQ$ touches the circle at points $P,Q$.
Prove that, $\angle PRQ+\angle PSQ={180}^{\circ }$

Answer 1

Given: Two circles intersect each other at points $S$ and $R$.
line $PQ$ is a common tangent.
To prove: $\angle PRQ+\angle PSQ={180}^{\circ }$
Proof:
Line $PQ$ is the tangent at point $P$ and seg $PR$ is a secant.
$\therefore [\angle RPQ=\angle PSR\dots \dots \dots$ (i)
and $\angle PQR=\angle QSR]$
(ii) [Tangent secant theorem]
In $△PQR$,
$\angle PQR+\angle PRQ+\angle RPQ={180}^{\circ }$ [Sum of the measures of angles of a triangle is ${180}^{\circ }]$
$\therefore \angle QSR+\angle PRQ+\angle PSR={180}^{\circ }[$ [rom (i) and (ii)]
$\therefore \angle PRQ+\angle QSR+\angle PSR={180}^{\circ }$
$\therefore \angle PRQ+\angle PSQ={180}^{\circ }[$ Angle addition property]